Properties

Label 3.331.4t5.2c1
Dimension 3
Group $S_4$
Conductor $ 331 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$331 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{3} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd
Determinant: 1.331.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 10\cdot 13 + 2\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 4\cdot 13^{6} + 2\cdot 13^{7} + 6\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 6 + 2\cdot 13 + 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{5} + 11\cdot 13^{6} + 12\cdot 13^{7} + 5\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 3 }$ $=$ $ a + 2 + \left(3 a + 9\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(10 a + 8\right)\cdot 13^{3} + \left(10 a + 8\right)\cdot 13^{4} + \left(6 a + 2\right)\cdot 13^{5} + \left(9 a + 8\right)\cdot 13^{6} + \left(8 a + 9\right)\cdot 13^{7} + \left(10 a + 5\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 12 a + 3 + \left(9 a + 11\right)\cdot 13 + \left(a + 3\right)\cdot 13^{2} + \left(2 a + 8\right)\cdot 13^{3} + \left(2 a + 8\right)\cdot 13^{4} + \left(6 a + 11\right)\cdot 13^{5} + \left(3 a + 10\right)\cdot 13^{6} + \left(4 a + 8\right)\cdot 13^{7} + \left(2 a + 7\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 3 + \left(3 a + 5\right)\cdot 13 + \left(4 a + 4\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(8 a + 1\right)\cdot 13^{4} + \left(10 a + 7\right)\cdot 13^{5} + \left(12 a + 7\right)\cdot 13^{6} + \left(6 a + 5\right)\cdot 13^{7} + \left(4 a + 1\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 5 a + 11 + 9 a\cdot 13 + \left(8 a + 5\right)\cdot 13^{2} + \left(3 a + 2\right)\cdot 13^{3} + \left(4 a + 1\right)\cdot 13^{4} + \left(2 a + 9\right)\cdot 13^{5} + 9\cdot 13^{6} + \left(6 a + 12\right)\cdot 13^{7} + \left(8 a + 11\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,6,4)(2,3,5)$
$(1,3,4)(2,6,5)$
$(3,5)(4,6)$
$(3,4)(5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$3$$2$$(1,2)(3,6)$$-1$
$6$$2$$(3,5)(4,6)$$1$
$8$$3$$(1,6,4)(2,3,5)$$0$
$6$$4$$(1,3,2,6)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.